Steidl, G. A note on the dual treatment of higher-order regularization functionals. (English) Zbl 1087.65067 Computing 76, No. 1-2, 135-148 (2006). Summary: We apply the dual approach developed by A. Chambolle [J. Math. Imaging Vision 20, No. 1–2, 89–97 (2004)] for the Rudin-Osher-Fatemi model to regularization functionals with higher order derivatives. We emphasize the linear algebra point of view by consequently using matrix-vector notation. Numerical examples demonstrate the differences between various second order regularization approaches. Cited in 37 Documents MSC: 65K10 Numerical optimization and variational techniques 49J20 Existence theories for optimal control problems involving partial differential equations 49M25 Discrete approximations in optimal control 49M29 Numerical methods involving duality Keywords:Rudin-Osher-Fatemi model; higher order regularization; convex optimization; dual optimization methods; G-norm; support vector regression; numerical examples Software:ftnonpar PDFBibTeX XMLCite \textit{G. Steidl}, Computing 76, No. 1--2, 135--148 (2006; Zbl 1087.65067) Full Text: DOI Link References: [9] Didas, S: Higher-order variational methods for noise removal in signals and images. Diplomarbeit, Universität des Saarlandes, 2003. [12] Hinterberger,W., Scherzer, O.: Variational methods on the space of functions of bounded Hessian for convexification and denoising. Technical Report, University of Innsbruck, Austria, 2003. · Zbl 1098.49022 [14] Lysaker,M., Lundervold, A.,Tai, X.-C.: Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time. Technical report CAM-02-44, Department of Mathematics, University of California at Los Angeles, 2002. · Zbl 1286.94020 [18] Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22. Providence: AMS 2001. · Zbl 0987.35003 [20] Obereder, A., Osher, S., Scherzer, O.: On the use of dual norms in bounded variation type regularization. Technical report, Department of Computer Science, University of Innsbruck, Austria, 2004. [25] Steidl, G., Didas, S., Neumann, J.: Relations between higher-order TV regularization and support vector regression. In: Scale-space and PDE methods in computer vision (Kimmel, R., Sochen, N., Weickert, J., eds.). Lecture Notes in Computer Science, vol. 3459. Berlin: Springer 2005, pp. 515–527. · Zbl 1119.68507 [28] Welk, M., Weickert, J., Steidl, G.: A four-pixel scheme for singular differential equations. In: Scale-space and PDE methods in computer vision (Kimmel, R., Sochen, N., Weickert, J., eds.). Lecture Notes in Computer Science, vol. 3459. Berlin: Springer 2005, pp. 610–621. · Zbl 1119.68512 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.